q -Hypergeometric and Related Functions - 17.12 Bailey Pairs

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
17.12.E1 ∑ n = 0 ∞ α n ⁢ γ n = ∑ n = 0 ∞ β n ⁢ δ n superscript subscript 𝑛 0 subscript 𝛼 𝑛 subscript 𝛾 𝑛 superscript subscript 𝑛 0 subscript 𝛽 𝑛 subscript 𝛿 𝑛 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\alpha_{n}\gamma_{n}=\sum_{n=0}% ^{\infty}\beta_{n}\delta_{n}}}
\sum_{n=0}^{\infty}\alpha_{n}\gamma_{n} = \sum_{n=0}^{\infty}\beta_{n}\delta_{n}		 

sum(alpha[n]*gamma[n], n = 0..infinity) = sum(beta[n]*delta[n], n = 0..infinity)
 
Sum[Subscript[\[Alpha], n]*Subscript[\[Gamma], n], {n, 0, Infinity}, GenerateConditions->None] == Sum[Subscript[\[Beta], n]*Subscript[\[Delta], n], {n, 0, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
17.12#Ex1 β n = ∑ j = 0 n α j ⁢ u n - j ⁢ v n + j subscript 𝛽 𝑛 superscript subscript 𝑗 0 𝑛 subscript 𝛼 𝑗 subscript 𝑢 𝑛 𝑗 subscript 𝑣 𝑛 𝑗 {\displaystyle{\displaystyle\beta_{n}=\sum_{j=0}^{n}\alpha_{j}u_{n-j}v_{n+j}}}
\beta_{n} = \sum_{j=0}^{n}\alpha_{j}u_{n-j}v_{n+j}		 

beta[n] = sum(alpha[j]*u[n - j]*v[n + j], j = 0..n)
 
Subscript[\[Beta], n] == Sum[Subscript[\[Alpha], j]*Subscript[u, n - j]*Subscript[v, n + j], {j, 0, n}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
17.12#Ex2 γ n = ∑ j = n ∞ δ j ⁢ u j - n ⁢ v j + n subscript 𝛾 𝑛 superscript subscript 𝑗 𝑛 subscript 𝛿 𝑗 subscript 𝑢 𝑗 𝑛 subscript 𝑣 𝑗 𝑛 {\displaystyle{\displaystyle\gamma_{n}=\sum_{j=n}^{\infty}\delta_{j}u_{j-n}v_{% j+n}}}
\gamma_{n} = \sum_{j=n}^{\infty}\delta_{j}u_{j-n}v_{j+n}		 

gamma[n] = sum(delta[j]*u[j - n]*v[j + n], j = n..infinity)
 
Subscript[\[Gamma], n] == Sum[Subscript[\[Delta], j]*Subscript[u, j - n]*Subscript[v, j + n], {j, n, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
17.12.E3 β n = ∑ j = 0 n α j ( q ; q ) n - j ⁢ ( a ⁢ q ; q ) n + j subscript 𝛽 𝑛 superscript subscript 𝑗 0 𝑛 subscript 𝛼 𝑗 q-Pochhammer-symbol 𝑞 𝑞 𝑛 𝑗 q-Pochhammer-symbol 𝑎 𝑞 𝑞 𝑛 𝑗 {\displaystyle{\displaystyle\beta_{n}=\sum_{j=0}^{n}\frac{\alpha_{j}}{\left(q;% q\right)_{n-j}\left(aq;q\right)_{n+j}}}}
\beta_{n} = \sum_{j=0}^{n}\frac{\alpha_{j}}{\qPochhammer{q}{q}{n-j}\qPochhammer{aq}{q}{n+j}}		 

beta[n] = sum((alpha[j])/(QPochhammer(q, q, n - j)*QPochhammer(a*q, q, n + j)), j = 0..n)

Subscript[\[Beta], n] == Sum[Divide[Subscript[\[Alpha], j],QPochhammer[q, q, n - j]*QPochhammer[a*q, q, n + j]], {j, 0, n}, GenerateConditions->None]
Failure Aborted Error
Failed [300 / 300]
Result: Complex[0.6508376433032488, -0.21856268949920582]
Test Values: {Rule[a, -1.5], Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[β, 1.5], Rule[Subscript[α, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[β, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.8660402331469415, 0.20457300495175623]
Test Values: {Rule[a, -1.5], Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[β, 1.5], Rule[Subscript[α, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[β, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
17.12.E4 ∑ n = 0 ∞ q n 2 ⁢ a n ⁢ β n = 1 ( a ⁢ q ; q ) ∞ ⁢ ∑ n = 0 ∞ q n 2 ⁢ a n ⁢ α n superscript subscript 𝑛 0 superscript 𝑞 superscript 𝑛 2 superscript 𝑎 𝑛 subscript 𝛽 𝑛 1 q-Pochhammer-symbol 𝑎 𝑞 𝑞 superscript subscript 𝑛 0 superscript 𝑞 superscript 𝑛 2 superscript 𝑎 𝑛 subscript 𝛼 𝑛 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}q^{n^{2}}a^{n}\beta_{n}=\frac{1% }{\left(aq;q\right)_{\infty}}\sum_{n=0}^{\infty}q^{n^{2}}a^{n}\alpha_{n}}}
\sum_{n=0}^{\infty}q^{n^{2}}a^{n}\beta_{n} = \frac{1}{\qPochhammer{aq}{q}{\infty}}\sum_{n=0}^{\infty}q^{n^{2}}a^{n}\alpha_{n}		 

sum((q)^((n)^(2))* (a)^(n)* beta[n], n = 0..infinity) = (1)/(QPochhammer(a*q, q, infinity))*sum((q)^((n)^(2))* (a)^(n)* alpha[n], n = 0..infinity)

Sum[(q)^((n)^(2))* (a)^(n)* Subscript[\[Beta], n], {n, 0, Infinity}, GenerateConditions->None] == Divide[1,QPochhammer[a*q, q, Infinity]]*Sum[(q)^((n)^(2))* (a)^(n)* Subscript[\[Alpha], n], {n, 0, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
17.12#Ex5 α n = ( a ; q ) n ⁢ ( 1 - a ⁢ q 2 ⁢ n ) ⁢ ( - 1 ) n ⁢ q n ⁢ ( 3 ⁢ n - 1 ) / 2 ⁢ a n ( q ; q ) n ⁢ ( 1 - a ) subscript 𝛼 𝑛 q-Pochhammer-symbol 𝑎 𝑞 𝑛 1 𝑎 superscript 𝑞 2 𝑛 superscript 1 𝑛 superscript 𝑞 𝑛 3 𝑛 1 2 superscript 𝑎 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 1 𝑎 {\displaystyle{\displaystyle\alpha_{n}=\frac{\left(a;q\right)_{n}(1-aq^{2n})(-% 1)^{n}q^{n(3n-1)/2}a^{n}}{\left(q;q\right)_{n}(1-a)}}}
\alpha_{n} = \frac{\qPochhammer{a}{q}{n}(1-aq^{2n})(-1)^{n}q^{n(3n-1)/2}a^{n}}{\qPochhammer{q}{q}{n}(1-a)}		 

alpha[n] = (QPochhammer(a, q, n)*(1 - a*(q)^(2*n))*(- 1)^(n)* (q)^(n*(3*n - 1)/2)* (a)^(n))/(QPochhammer(q, q, n)*(1 - a))

Subscript[\[Alpha], n] == Divide[QPochhammer[a, q, n]*(1 - a*(q)^(2*n))*(- 1)^(n)* (q)^(n*(3*n - 1)/2)* (a)^(n),QPochhammer[q, q, n]*(1 - a)]
Error Failure -
Failed [300 / 300]
Result: Complex[5.814582562299427, -3.4240381056766607]
Test Values: {Rule[a, -1.5], Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[Subscript[α, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-12.010896071760529, -4.7481964481437355]
Test Values: {Rule[a, -1.5], Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[Subscript[α, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
17.12#Ex6 β n = 1 ( q ; q ) n subscript 𝛽 𝑛 1 q-Pochhammer-symbol 𝑞 𝑞 𝑛 {\displaystyle{\displaystyle\beta_{n}=\frac{1}{\left(q;q\right)_{n}}}}
\beta_{n} = \frac{1}{\qPochhammer{q}{q}{n}}		 

beta[n] = (1)/(QPochhammer(q, q, n))

Subscript[\[Beta], n] == Divide[1,QPochhammer[q, q, n]]
Failure Failure Error
Failed [297 / 300]
Result: Complex[0.3660254037844387, -1.366025403784439]
Test Values: {Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[β, 1.5], Rule[Subscript[β, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[2.232050807568878, -0.8660254037844388]
Test Values: {Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[β, 1.5], Rule[Subscript[β, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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